Application Example 12
(Functions of random variables and reliability analysis)
DISTRIBUTION OF WAVES AND WAVE LOADS IN A
RANDOM SEA
The Ocean Surface and the Distribution of Wave Height
The elevation of the sea surface during a storm is adequately described by what is called
a normal (or Gaussian) random function. This means that, at a generic location and time,
the sea surface elevation has with good approximation a normal distribution around the
mean sea elevation. Considerations from the theory of normal random functions lead to a
distribution of wave height H of the Rayleigh type, with CDF
FH (h) = 1− e
−2(h / Hs ) 2
(1)
where the parameter HS, called the significant wave height, is four times the standard
deviation of the sea surface elevation at a generic point. H in Eq. 1 has mean value 0.627
HS and standard deviation 0.328 HS.
Curve (a) in Figure 1 is a plot of the Rayleigh probability density fX (x) = xe−x
2
/2
of the normalized variable X = 2H /H s . Our main interest is in the upper tail of the
distribution of H, because the larger waves are those that pose threat to ships, offshore
structures and marine operations in general.
Maximum Wave Height in a Storm
FH(h) in Eq. 1 is the probability that a wave chosen at random in a sea state with
parameter HS has height less than h. Consider now a storm with 1000 waves (this number
of waves is representative of a storm having an effective duration of 3 hours and an
average wave period of 10 seconds) and suppose that such waves have independent and
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identically distributed heights. Then the distribution of the maximum wave height during
the storm, say H1000, is
FH1000 (h) = {FH (h)}
1000
1000
⎧
−2( h / H s )2 ⎫
⎬
= ⎨1 − e
⎭
⎩
(2)
The probability density function of X1000 = 2H 1000 /H s is shown in Figure 1, as curve
(b). For 1000 waves, the most probable maximum height is found to be 1.868HS. The
distribution of maximum wave height is not very sensitive to the storm duration. For
example, for a storm with 2000 waves, the most probable maximum wave height is
1.958HS; see curve (c) in Figure 1, which shows the probability density function of
X 2000 = 2H 2000 /H s .
Maximum Wave Load During a Single Storm
For engineering design or safety verification, one is concerned not just with the
distribution of wave height, but also with the distribution of the resulting load on the
structure of interest. A simple model for the load q applied by a wave of height h to a
fixed platform is
q ∝ hC
(3)
where C is a constant whose value is close to 2. Therefore, if one denotes by h0 the
design wave height and by q0 the associated design force, the relation between q and h is
q = q0 (h/h0)C
h = h0 (q/q0)1/C
(4)
Since the relationship between q and h is monotonically increasing, the distributions of
wave height H and load Q are related as
2
FQ (q) = FH [h(q)]
[
1/C
= FH h o (q / qo )
]
(5)
Using for FH the distribution in Eq. 1 or 2, one obtains the distribution of the maximum
load Q due to one or 1000 waves, in a sea with significant wave height HS. These load
distributions are
FQ (q) = 1 − e
FQ
1000
−2(h o / Hs )2 (q / q o )2 / C
{
(6a)
}1000
(q) = FQ (q)
(6b)
1000
⎧
−2( h o / H s ) 2 (q / q o ) 2 / C ⎫
= ⎨1 − e
⎬
⎩
⎭
A plot of the exceedance probability P[Q > q] = e−2(h o / H s )
2
(q /q o ) 2 / C
against q/qo for a
single wave is shown as curve (a) in Figure 2, where it has been assumed that C = 2 and
h0 = 1.868HS (as was said before, this value of ho corresponds to the most probable wave
height in a 1000-wave storm with significant wave height HS).
{
probability P[Q1000 > q] = 1− 1− e
−2(h o / H s ) 2 (q /q o ) 2 / C
}
The exceedance
1000
during a 1000-wave storm is
plotted as curve (b). As one would expect, curve (b) shows that there is a probability
about 50% that the design load is exceeded during a 1000-wave storm with the value of
HS used for design. Curve (b) also shows that during the “design storm” there is a
probability 0.05 that the maximum load on the structure will exceed the design load by
40%.
Maximum Annual Load
Finally, consider the distribution of the maximum annual load Qyear on a platform at a
certain site, say in the Gulf of Mexico. The main threat to the platform comes from
hurricanes. In order to obtain the distribution of the annual maximum HS at the site,
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HSmax, one typically uses a simulation approach in which synthetic hurricane occurrences
and their paths are numerically generated together with their intensity, size and speed
parameters. A sea state model is then used to predict the temporal evolution of HS at the
site for each simulated hurricane. From simulations covering a long period of time, the
distribution of HSmax is then obtained. For example, Figure 3 shows the distribution of
HSmax derived from 1600 years of simulation at a site south of Galveston, in 600 feet of
water. A distribution of the following (truncated exponential) type fits the empirical data
well:
1− FH s max (h) = 0.47,
for 0 < h ≤ 4.14m
= 3.11e−h / 2.19 ,
for h > 4.14m
(7)
Notice that 1 – 0.47 = 0.53 is the probability that HSmax = 0, i.e. the probability that, in a
generic year, the site does not experience hurricanes.
Problem 12.1
(a) Calculate the mean value of Hsmax in Eq. 7, E[HSmax].
(b) Combine the distribution of Q1000 for given HS in Eq. 6b for C = 2 and h0 = 3E[HSmax]
with the distribution of HSmax in Eq. 7 to obtain the distribution of the maximum
normalized load Q1000/q0 in one year (assuming that the maximum load occurs during
the hurricane with significant wave height HSmax).
What is the probability of
exceeding the design load q0 in one year?
(c) Suppose that the safety goal is to design the platform for a load that is exceeded with
probability 0.01 in one year. Modify the above design wave height h0 to achieve this
safety goal.
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